Three metallic spheres with radii 3 cm,4 cm, and 5 cm respectively are melted together and recast into a single solid sphere. What is the radius of the new sphere?

Correct — C. Melting keeps the volume, and (4⁄3)π is common to every sphere, so add the cubes of the radii.

Sum of cubes = 3³ + 4³ + 5³
= 27 + 64 + 125 = 216

New radius: R³ = 216, so R = ∛216 = 6 cm → option (c)

Why the others are wrong

• (a)
  12 cm — 12 = 3 + 4 + 5 adds the radii. Volume adds as the cube: 12³ = 1,728, eight times the 216 available.
• (b)
  9 cm — 9³ = 729, more than three times the 216 that 3³ + 4³ + 5³ supplies.
• (d)
  7 cm — 7³ = 343, well above 216. Adding squares instead of cubes gives √(9 + 16 + 25) = √50 ≈ 7.07, close to this option.

When solids are melted and recast, volume is conserved. Surface area is not.

For spheres V = (4⁄3)πr³, and (4⁄3)π cancels from both sides, leaving R³ = r₁³ + r₂³ + r₃³.

3³ + 4³ + 5³ = 6³ is an identity worth knowing. Contrast the Pythagorean 3² + 4² = 5², which is about squares.

Key facts

• Volume of a sphere = (4⁄3)πr³.
• Recasting spheres into one: R³ = r₁³ + r₂³ + r₃³.
• 3³ + 4³ + 5³ = 216 = 6³.

Common traps

• Adding the radii, which gives 12 cm.
• Adding squares of the radii as if surface area were kept, which gives about 7.07 cm.

No directly related past PYQ was found.